Finite incidence, recurrent response, and physical access

This chapter provides a finite example of why the provenance of a physical structure matters in addition to its spectrum. The calculations retain which parts of a carrier arose from a designated source region and which were added. They illustrate the realization discipline used later in $R^\ast$: an invariant numerical summary need not retain every distinction relevant to a proposed physical or phenomenal target.

12.1 Source-sensitive response carriers

A finite source-response construction begins with a rooted partial order, not an empirically fitted spectrum. The retained root is $0<1$, and the fresh loci are $2,3$. The order complex contains every nonempty chain, including compositional higher simplices. With a counting-normalized orthonormal simplex basis, the signed boundary operator $d$ obeys $d^2=0$. Define

This is finite Hodge mathematics. Its source status depends on the admitted response principles and Hilbertization, as the constructive O1 source explicitly states [39].

Separate chains lying entirely in the root support from mixed-interface and fresh chains. Let $P_R$ project onto root-supported chains and $P_F=I-P_R$ onto the remainder. The decomposition is assigned by provenance before the spectral calculation. It is not chosen to produce a desired gap.

For an eigenprojection $E_\lambda$ of $K$, define $B_\lambda=P_RE_\lambda P_F$. The positive RSM-active spectrum is

This differs from the ordinary spectral gap because it depends on the geometry of the retained/fresh sectors. A zero-eigenvalue cross block may exist while being excluded by the word “positive” in this definition.

Theorem 12.1 (Finite return identity)

For finite $s,t\ge0$,

$$P_Fe^{-sK}E_\lambda P_RE_\lambda e^{-tK}P_F =e^{-\lambda(s+t)}B_\lambda^\dagger B_\lambda.$$

It is positive and nonzero exactly when $B_\lambda\ne0$.

Proof

Since $E_\lambda$ is the whole orthogonal eigenprojection, $e^{-tK}E_\lambda=e^{-t\lambda}E_\lambda$. Move only these commuting factors, not $P_R$, through the exponentials. The remaining product is $P_FE_\lambda P_RE_\lambda P_F=B_\lambda^\dagger B_\lambda$. A matrix $B^\dagger B$ vanishes exactly when $B=0$.

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Degenerate eigenspaces cause no ambiguity because an arbitrary eigenvector is not substituted for $E_\lambda$. In an infinite-dimensional model the same identity applies to a genuine finite-eigenvalue spectral atom, but singleton projectors can miss the entire continuous spectrum. The physical interpretation also requires an instrument realizing the intervening operations and a retained record. An operator product is not, by itself, a physical recursive experiment.

12.2 Five exact profiles

The source relations and independently recomputed results are shown in Table 12.1. The common relation $0<1$ is implicit in each row.

Profile	Additional relations	$\dim\HH$	Spectrum of $K$
$V2_{02}$	$0<2,\ 0<3$	7	$0^1,1^4,4^2$
$V2_{34}$	$2<1,\ 3<1$	7	$0^1,1^4,4^2$
$V2_{04}$	$0<2,0<3,1<2,1<3$	11	$0^1,2^4,4^6$
$V2_{11}$	$0<2,0<3,2<1,3<1$	11	$0^1,2^4,4^6$
$V2_{29}$	$2<0,2<1,3<0,3<1$	11	$0^1,2^4,4^6$

In every case $P_R$ has rank three, supported by the two root vertices and root edge. For the seven-dimensional profiles, $\rank B_0=1$, $\rank B_1=2$, and $\rank B_4=2$. For the eleven-dimensional profiles, the corresponding ranks are $1,0,1$ at eigenvalues $0,2,4$. The positive active gap is therefore one in the former pair and four in the latter three.

For exact calculation, the eigenprojectors can be obtained without choosing eigenvector bases:

The verification package supplies all boundary matrices, $D,K,P_R$, projectors, and cross ranks. It checks all 24 signed vertex relabellings for each of the five profiles. A relabelling must transport the root support and simplex orientation; an unsigned permutation that changes orientation is not a valid matrix comparison.

Equal spectra do not identify source objects. Different differentials, provenance projections, and admitted interactions can share $K$ or its spectrum. Even for a fixed diagonal $K$, choosing a commuting projector gives no cross block, while a rotated projector can give one. The calculation therefore supports a source-sensitive response classification, not a universal consciousness threshold.

12.3 Interaction-relative sector invariance

Let $P=P_R$, $J=i[K,P]$, and in the eleven-dimensional examples set $\Pi=J^2/4$. Exact calculation gives

The block $\Pi D(I-\Pi)$ has rank one and singular value two. Thus a $D$-generated actuator would mix part of the two support sectors while conserving $K$, but an action algebra generated by $K,P,J$ preserves them.

The $P$-commutation has a general explanation. Relative to $P\HH\oplus(I-P)\HH$, write

Then $J^2=\diag(BB^\dagger,B^\dagger B)$, so $P$ commutes with $J^2$ and its support. The additional condition needed for the full $K/P/J$ obstruction is preservation of that support by $K$. In the specified profiles it holds.

If every Kraus operator of an admitted instrument commutes with $\Pi$, an input supported in one extreme sector remains there on every nonzero conditional branch. Such a trace-preserving channel preserves $\tr\rho\Pi$ unconditionally. Conditioning a mixed-sector input can change its normalized sector weight, but cannot create support absent from an extreme input. This is an interaction-relative conservation law, not a fundamental superselection of the whole matrix algebra.

12.4 Actuation, source drift, and seed-specific mobility

Mathematical membership of $D$ in an operator algebra does not expose a $D$ port or supply a Hamiltonian coupling. Even the conditions of self-adjointness, source covariance, and $K$ conservation leave a family $h(K)D$. Selecting the primitive $D$ as an actuator is a possible realization law, not a consequence of symmetry alone.

For an actual Hermitian actuator $A$ and state $\rho=\Pi\rho\Pi$, leakage has the expansion

The first derivative vanishes. A nonzero commutator guarantees some possible mixing, not first-order escape of every extreme seed. In the type-11 crossing channel there are unit vectors $u\in\im\Pi$, $v\in\ker\Pi$ with $Du=2v$, $Dv=2u$, hence $e^{-itD}u=\cos(2t)u-i\sin(2t)v$. An additional active zero-mode line is $D$-dark. These facts are about reachability in a supplied action grammar.

Similarly, an isolated tap generated by $\eta J\otimes X$ and a tap with continuing source drift are different operations. In the active two-level sector, the latter has, after a scalar shift,

Its memory-one effect is

It is not a perfect $\Pi$ measurement at finite $\eta$. Switching off or refocusing drift requires an admitted control and resources. An imperfect nontrivial tap can still provide useful conditional information, but a family of ensemble evaluations is not exact probability estimation from a single unknown specimen.

12.5 Robust spectral access

Let a contour $\Gamma$ isolate a spectral cluster of Hermitian $K$ at distance $a>0$ from its spectrum. For $K'=K+\Delta K$, $\norm{\Delta K}=\varepsilon<a$, the Riesz projections obey

The resolvent identity gives an integrand norm at most $\varepsilon/[a(a-\varepsilon)]$, and integration proves the bound. Write $E=P_\Gamma(K)$ and $E'=P_\Gamma(K')$ for these spectral projections. If the provenance projector also changes from $P$ to $P'$ by $\delta$, then

A cross block larger than this uncertainty remains nonzero; rank claims need further control. Spectral robustness is not evidence of awareness.